The present invention relates to fiber non-linearity compensation.
Fiber-based amplifiers offer the ability to amplify ultrafast pulses to energies comparable with conventional bulk solid-state systems with significant practical advantages such as compactness, reduction of complex components, and freedom from misalignment. However, the smaller beam confinement and larger interaction lengths render them vulnerable to nonlinear effects, for single wavelength transmission (compared with WDM case), the dominant of which is self-phase modulation (SPM). Due to the Kerr effect, high optical intensity in a medium (e.g. an optical fiber) causes a nonlinear phase delay which has the same temporal shape as the optical intensity. This can be described as a nonlinear change in the refractive index:Δn=n2Iwith the nonlinear index n2 and the optical intensity I. In the context of self-phase modulation, the emphasis is on the temporal dependence of the phase shift, whereas the transverse dependence for some beam profile leads to the phenomenon of self-focusing.
Although the refractive index is a very weak function of signal power, the higher power from optical amplifiers and long transmission distances make it no longer negligible in modern optical communication systems. In fact, phase modulation distortion due to intensity dependent refractive index induces various nonlinear effects, namely, self-phase modulation (SPM) and cross-phase modulation (XPM). (Four-wave mixing (FWM) is another non-linearity distortion but not related to refractive index.)
One nonlinear phase shift originating from the Kerr effect is cross-phase modulation (XPM). While SPM is the effect of a pulse on it own phase, XPM is a nonlinear phase effect due to optical pulses in other channels. Therefore, XPM occurs only in multi-channel systems. In a multi-channel system, the nonlinear phase shift of the signal at the center wavelength λi is described as,
      ϕ    NL    =                    2        ⁢        π                    λ        i              ⁢          n      2        ⁢          z      [                                    I            i                    ⁡                      (            t            )                          +                  2          ⁢                                    ∑                              i                ≠                j                                      ⁢                                          I                j                            ⁡                              (                t                )                                                        ]      
The first term is responsible for SPM, and the second term is for XPM. The above equation might lead to a speculation that the effect of XPM could be at least twice as significant as that of SPM. However, XPM is more effective when pulses in the other channels are synchronized with the signal of interest. When pulses in each channel travel at different group velocities due to dispersion, the pulses slide past each other while propagating. FIG. 1A illustrates how two isolated pulses in different channels collide with each other. When the faster traveling pulse has completely walked through the slower traveling pulse, the XPM effect becomes weaker. The relative transmission distance for two pulses in different channels to collide with each other is called the walk-off distance.
      L    w    =                    T        o                                                                v              g                              -                1                                      ⁡                          (                              λ                1                            )                                -                                    v              g                              -                1                                      ⁡                          (                              λ                2                            )                                                    ≈                  T        o                                      D          ⁢                                          ⁢          Δλ                            where To is the pulse width, vg is the group velocity, and λ1, λ2 are the center wavelength of the two channels. D is the dispersion coefficient, and Δλ=|λ1−λ2|.
When dispersion is significant, the walk-off distance is relatively short, and the interaction between the pulses will not be significant, which leads to a reduced effect of XPM. However, the spectrum broadened due to XPM will induce more significant distortion of temporal shape of the pulse when large dispersion is present, which makes the effect of dispersion on XPM complicated.
The dependence of the refractive index on optical intensity causes a nonlinear phase shift while propagating through an optical fiber. The nonlinear phase shift is given by
      ϕ    NL    =                    2        ⁢        π            λ        ⁢          n      2        ⁢          I      ⁡              (        t        )              ⁢    z  where λ is the wavelength of the optical wave, and z is the propagation distance.
Since the nonlinear phase shift is dependent on its own pulse shape, it is called self-phase modulation (SPM). When the optical signal is time varying, such as an intensity modulated signal, the time-varying nonlinear phase shift results in a broadened spectrum of the optical signal. If the spectrum broadening is significant, it may cause cross talk between neighboring channels in a dense wavelength division multiplexing (DWDM) system. Even in a single channel system, the broadened spectrum could cause a significant temporal broadening of optical pulses in the presence of chromatic dispersion.
Back-propagation method has been proposed to compensate the fiber non-linearity. The NLSE is an invertible equation. In the absence of noise, the transmitted signal can be exactly recovered by “back-propagating” the received signal through the inverse NLSE given by:
            ∂      E              ∂      z        =            (                        -                      D            ^                          -                  N          ^                    )        ⁢    E  This operation is equivalent to passing the received signal through a fictitious fiber having opposite-signed parameters, such as through a receiver side back propagation 10 (FIG. 1A). It is also possible to perform back-propagation at the transmitter side by pre-distorting the signal to invert the channel, and then transmitting the pre-distorted waveform through a transmitter side back propagation 12 (FIG. 1B). In the absence of noise, both schemes are equivalent.
Back-propagation operates directly on the complex-valued field E(z,t). Hence, the technique is universal, as the transmitted signal can have any modulation format or pulse shape, including multicarrier transmission using OFDM.
Some differences between optical system simulation and impairment compensation may occur. In the former, knowing the input to a fiber enables the output be computed to arbitrary precision; whereas in back-propagation, noise prevents exact recovery of the transmitted signal. It has been demonstrated that in the presence of noise, a modified back-propagation equation is effective in compensating nonlinearity:EBP(z,t)=exp(−h({circumflex over (D)}+ξ{circumflex over (N)}))EBP(z+h,t),where 0≦ξ≦1 is the fraction of the nonlinearity compensated. For every set of system parameters, there exists an optimum ξ that minimizes the mean square error (MSE) between the transmitted signal E(0,t) and the back-propagation solution EBP(0, t). In zero-dispersion fiber, for example, where back-propagation is equivalent to nonlinear phase rotation, it was shown that ξ=0.5 is optimal.
The existence of an optimum ξ can be appreciated by considering that in a typical fiber, the magnitude of the dispersion operator is much greater than the nonlinear operator. Thus, nonlinearity can be viewed as a perturbation to a mostly dispersive channel. The optimum phase to de-rotate at each back-propagation step depends on the accuracy of EBP(z, t) as an estimate of E(z,t). The more accurately the receiver estimates E(z,t), the closer ξ can be set to one, since the nonlinear phase rotation will lead to an output closer to the original signal. Conversely, if E(z,t) is not known accurately, error in amplitude will be converted to random phase rotations by the nonlinear operator hξ{umlaut over (N)}, yielding an output that is even further away from the desired signal in Euclidean distance. Hence, the optimum ξ depends on the received SNR as well as any uncompensated distortions that are present during back-propagation.
The receiver shown in FIG. 2 has been proposed for single-carrier transmission system with a coherent optical to electrical conversion system 20. System 20 includes a polarizing beam splitter (PBS) 22 and two 90-degree hybrids coupled to one or more analog to digital converters 24. In system 20, a linear equalizer (FSE) 28 follows a back-propagation module 26. In the absence of nonlinearity, back-propagation function inverts the fiber CD, so PMD is mitigated by the linear equalizer. At realistic transmission distances and symbol rates, PMD has only short duration, so we expect the signal amplitude profile will not be significantly distorted by PMD. Hence, back-propagation with the linear operator can still compensate most of the interactions between CD and nonlinearity. The linear equalizer compensates PMD and any residual linear effects not already compensated by back-propagation. If back-propagation includes PMD, the linear equalizer is reduced to a fixed down-sampler.
The ability of back-propagation to undo nonlinear effects depends on how accurately it can estimate the signal amplitude profile at every point in the fiber. Noise, PMD, and other distortions not estimated by the receiver, but which change the signal intensity profile, thus degrading performance. Since these effects accumulate with distance, the further a signal is back-propagated, the higher the relative error. In receiver-side back-propagation, the signal intensity profile is known accurately at the receiver, but becomes progressively less accurate as it is traced back to the transmitter.
FIG. 3A shows an exemplary arrangement where back-propagation can be done either at the transmitter or receiver side, or can also been split between the transmitter and receiver: transmit-side back-propagation inverts the first half of the channel, while receive-side back-propagation inverts the second half. In FIG. 3A, input data is provided to a back-propagation non-linearity compensation module 40, whose output is provided to an array of digital to analog converters 42. The analog data is provided to an array of electrical-to-optical upconverters 44 and sent to an arrayed waveguide grating (AWG) 46 for transmission to another AWG 48. At AWG 48, the information is converted using optical-to-electrical converters 50 and provided to an array of analog to digital converters 52 whose outputs are provided to a back-propagation non-linearity compensation module 54. Module 54 in turn is connected to an array of linear equalizers 56 driving an array of carrier recovery circuits 58 that generate output data. Since the XPM happens between different channels, multiple channel inputs and outputs need to be processed jointly with the non-linearity compensation module or processor 54 to remove the dispersion caused by XPM during the transmission.
FIG. 3B shows another exemplary arrangement with back-propagation at a transmitter 70 and a receiver 90. At the transmitter 70, data input is provided to an OFDM modulator 72 driving a back-propagation module 74, whose output is applied to a digital to analog converter 76 and provided to an E-O up-converter 78 and transmitted over an optical cable 80 to the receiver 90. At the receiver 90, an O-E down converter 92 receives the data which is provided to an analog to digital converter 94. The digital data is provided to a back-propagation module 96 and optionally to an OFDM demodulator 98. The data is provided to a linear equalizer 100 and then presented to a carrier recovery circuit 102 to generate output data. In FIG. 3B, the back-propagation is split evenly between the transmitter and receiver: transmit-side back-propagation inverts the first half of the channel, while receive-side back-propagation inverts the second half. To account for the change in relative error with distance, the parameter ξ should also vary with distance; a larger ξ is used for the spans closer to the transmitter (and receiver), while a smaller ξ is used for spans further away, where the estimated signal intensity is less reliable.
One challenge for commercial implementation of the non-linearity compensation process is the high computing complexity. If the transmitter or receiver side non-linearity compensation is used, the (back-propagation) non-linearity compensation function has to run in a real-time mode with multiple steps to compensate the linear and non-linear dispersion span by span. Even with the recent efforts in process simplification, the computing complexity of non-linearity compensation is still two orders of magnitude (greater 50 times) greater than the computing complexity of the linear dispersion compensation (1-tap frequency domain equalization) of the same transmission range.